\chapter{A 2-Stage Linear SP Approach to Design of Networks Under Uncertainty}

\label{chap-four}
In chapter \ref{chap-two} we introduced a survivable network desion/expansio model which assumes demands and links availability are affected by several failure scenarios the network faces. The decision maker needed to choose a set of capacity expansion alternatives for links in order to meet the demands under all the failure scenarios while minimizing the cost of design. However, in many service network design problems the decision maker may be willing to tolerate a certain amount of unmet demand at a cost. In such cases, the decision making becomes more complicated because of the fact that the amount of unmet demand and the inherent penalties are dependent of problem's uncertain parameters. For this reason the decision maker may want to consider the expected value of the inherent costs of capacity design and unmet demands. Our study is motivated by problems faced in protecting critical infrastructures including transportation network. To be completed
\section{Problem Statement}
\section{Model Formulation}
\section{Structural Properties}
\section{Solution Methodology}



\subsection {Tentative : Implement an alternate modeling approach : Minimum cost capacity design and expansion in failure prone networks}




\paragraph{Assumption 1}
\textit{ The set of realizations of the random variable $\xi$ is finite and it follows a discrete probability distribution, i.e. it has a predetermined number of possible values $\xi \in\left\{\xi^1,\xi^2,...,\xi^{|S|}\right\}$. The possible realizations of $\xi$ correspond to the occurrence of failure scenarios $s_{1},s_{2},...,s_{|S|}$ respectively.}


